The deformed Hermitian-Yang-Mills (dHYM) equation is the mirror equation for the special Lagrangian equation. The “small radius limit” of the dHYM equation is the J-equation, which is closely related to the constant scalar curvature K"ahler (cscK) metrics. In this talk, I will explain my recent result that the solvability of the J-equation is equivalent to a notion of stability.
I will also explain my similar result on the supercritical dHYM equation.